DOUBLING CONDITIONS FOR HARMONIC MEASURE IN JOHN
DOMAINS
HIROAKI AIKAWA AND KENTARO HIRATA
Abstract. We introduce new classes of domains, i.e., semi-uniform domains and inner
semi-uniform domains. Both of them are intermediate between the class of John domains
and the class of uniform domains. Under the capacity density condition, we show that the
harmonic measure of a John domain D satisfies certain doubling conditions if and only if
D is a semi-uniform domain or an inner semi-uniform domain.
1. Introduction
Let D be a bounded domain in Rn with n ≥ 2, δD (x) = dist(x, ∂D) and x0 ∈ D. Let
us recall some nonsmooth domains. By the symbol A, we denote an absolute positive
constant whose value is unimportant and may change from line to line. If necessary, we
use A0 , A1 , . . . , to specify them. We shall say that two positive functions f1 and f2 are
comparable, written f1 ≈ f2 , if and only if there exists a constant A ≥ 1 such that A−1 f1 ≤
f2 ≤ A f1 . The constant A will be called the constant of comparison. We write B(x, R) and
S (x, R) for the open ball and the sphere of center at x and radius R, respectively.
We say that D is a John domain with John constant c J > 0 and John center x0 ∈ D if
each x ∈ D can be joined to x0 by a rectifiable curve γ ⊂ D such that
(1)
δD (y) ≥ c J ℓ(γ(x, y)) for all y ∈ γ,
where γ(x, y) and ℓ(γ(x, y)) stand for the subarc of γ connecting x and y and its length,
respectively. In general, 0 < c J < 1. We say that D is a uniform domain if there exists a
constant A > 1 such that each pair of points x, y ∈ D can be joined by a rectifiable curve
γ ⊂ D such that ℓ(γ) ≤ A|x − y| and
(2)
min{ℓ(γ(x, z)), ℓ(γ(z, y))} ≤ AδD (z)
for all z ∈ γ.
We call this curve γ a cigar curve connecting x and y. See [11, 12, 15]. If the complement
of a uniform domain D satisfies the corkscrew condition, then D becomes an NTA domain
([13]). Observe that connectivity of a uniform domain can be extended from x, y ∈ D to
x, y ∈ D. We introduce the following class of domains.
Definition 1. We say that D is a semi-uniform domain if every pair of points x ∈ D and
y ∈ ∂D can be joined by a rectifiable curve γ such that γ \ {y} ⊂ D, ℓ(γ) ≤ A|x − y| and (2)
holds.
2000 Mathematics Subject Classification. 31B05, 31B25, 31C35.
Key words and phrases. John domain, semi-uniform domain, inner semi-uniform domain, harmonic
measure, doubling condition, capacity density condition.
This work was supported in part by Grant-in-Aid for Scientific Research (B) (No. 15340046) Japan
Society for the Promotion of Science.
1
2
HIROAKI AIKAWA AND KENTARO HIRATA
A Denjoy domain is a typical semi-uniform domain which is not necessarily uniform.
The relationships among above domains are summarized as
(3)
NTA $ Uniform $ Semi-uniform $ John.
Let ω(x, E, U) be the harmonic measure of the set E in an open set U evaluated at
x. Jerison-Kenig [13] proved that harmonic measure of an NTA domain D satisfies the
strong doubling condition: there is a constant A0 > 2 such that
(4)
ω(x, B(ξ, 2R) ∩ ∂D, D) ≤ Aω(x, B(ξ, R) ∩ ∂D, D) for x ∈ D \ B(ξ, A0 R),
where ξ ∈ ∂D and R > 0 small, say R ≤ RS D . If (4) holds only for some fixed point x = x0 ,
we say that the harmonic measure of D satisfies the doubling condition. Obviously the
strong doubling condition implies the doubling condition. Moreover, they showed that a
bounded planar simply connected domain D is an NTA domain if and only if the harmonic
c
measures both for D and D satisfy the doubling condition ([13, Theorem 2.7]). Kim and
Langmeyer [14] gave the one-sided analogue; a bounded planar Jordan domain is a John
domain if and only if the harmonic measure only for D satisfies the doubling condition.
Their argument is based on complex analysis as well.
Balogh-Volberg [6, 7] showed a doubling condition similar to (4) in a planar uniformly
John domain, or inner uniform domain (see Definition 2 below and the remarks before it).
They also pointed out that there is a planar inner uniform domain for which (4) fails to
hold. Indeed, let D be the complement of the line segments [−1, 1] and Lθ = {te−iθ : 0 ≤
t ≤ 1} with 0 < θ < π/2. Let B1 = B(te−iθ , ct) and B2 = B(te−iθ , 2ct), where 12 sin θ < c <
sin θ. Since B1 ∩ [−1, 1] = ∅ and B2 ∩ [−1, 1] , ∅, we have ω(x0 , B1 ∩ ∂D, D) ≈ tπ/(π−θ)
and ω(x0 , B2 ∩ ∂D, D) ≈ t as t → 0. Hence ω(x0 , B2 ∩ ∂D, D)/ω(x0 , B1 ∩ ∂D, D) → ∞.
See Figure 1.
B2
B1
Figure 1. Harmonic measure fails to satisfy the doubling condition.
In this paper, we characterize John domains whose harmonic measure satisfies (4), the
strong doubling condition. There is a John domain with polar boundary whose harmonic
measure vanishes. For such domains any doubling conditions for harmonic measure is
hopeless. To avoid such pathological domains, we assume the capacity density condition
(abbreviated to CDC). See Section 3 for its definition. If n = 2, then the CDC coincides
with the uniform perfectness of the boundary. Our main result is as follows.
Theorem 1. Let D be a John domain with John constant c J and suppose the CDC holds.
Then the following are equivalent:
(i) D is a semi-uniform domain.
(ii) The harmonic measure of D satisfies the strong doubling condition, i.e., (4) holds
whenever ξ ∈ ∂D and R > 0 is small.
DOUBLING CONDITIONS FOR HARMONIC MEASURE
3
(iii) For each α > 1/c J , there exist constants A > 1 and τ > 0 depending only on D
and α such that
(
)τ
1
R
(5)
ω(x, ∂D ∩ B(ξ, R), D) ≥
for |x − ξ| < αδD (x),
A R + |x − ξ|
whenever ξ ∈ ∂D and R > 0 is small.
Remark 1. The constant 1/c J is a threshold; if α is less than c J , then {x ∈ D : |x − ξ| <
αδD (x)} may be an empty set.
Next, we state a version of Theorem 1 with respect to the inner diameter metric ρD (x, y)
defined by
ρD (x, y) = inf{diam(γ) : γ is a curve connecting x and y in D},
where diam(γ) denotes the diameter of γ. If we replace diam(γ) by ℓ(γ) in the above
definition, then we obtain the inner length distance λD (x, y). Obviously |x−y| ≤ ρD (x, y) ≤
λD (x, y). It turns out, however, that ρD and λD are comparable for a John domain (Väisälä
[16, Theorem 3.4]). We say that D is an inner uniform domain or uniformly John domain
if there exists a constant A > 1 such that every pair of points x, y ∈ D can be connected
by a curve γ ⊂ D with ℓ(γ) ≤ AρD (x, y) and (2). See Balogh-Volberg [6, 7] and BonkHeinonen-Koskela [9]; actually, the latter use λD (x, y) instead of ρD (x, y) in the definition.
However, ρD and λD are equivalent as noted above. For a John domain D, we can consider
the completion D∗ with respect to ρD ([4, Proposition 2.1]). Then ∂∗ D = D∗ \ D is the
ideal boundary of D with respect to ρD . Observe that connectivity of an inner uniform
domain can be extended from x, y ∈ D to x, y ∈ D∗ . See [4, Lemma 2.1].
Definition 2. We say that D is an inner semi-uniform domain if every pair of points x ∈ D
and y ∈ ∂∗ D can be joined by a rectifiable curve γ such that γ \ {y} ⊂ D, ℓ(γ) ≤ AρD (x, y)
and (2) holds.
Let ξ∗ ∈ ∂∗ D. Then there are a point ξ ∈ ∂D and a sequence {x j } ⊂ D converging to ξ
with respect to the Euclidean metric as well as converging to ξ∗ with respect to ρD . We
say that ξ∗ lies over ξ and define the projection π from D∗ to D by π(ξ∗ ) = ξ for ξ∗ ∈ ∂∗ D
and π|D = id |D . Let Bρ (ξ, R) be the connected component of B(ξ, R) ∩ D from which
ξ∗ is accessible. We observe that Bρ (ξ, R) plays a role of a ball with center at ξ∗ in the
completion D∗ ([4, Lemma 2.2]). Let ∆ρ (ξ∗ , R) = {x∗ ∈ ∂∗ D : ρD (x∗ , ξ∗ ) < R}. This is a
surface ball with respect to ρD . Consider a version of (4) with respect to ρD : there is a
constant A0 > 2 such that
(6)
ω(x, ∆ρ (ξ∗ , 2R), D) ≤ Aω(x, ∆ρ (ξ∗ , R), D) for x ∈ D \ Bρ (ξ∗ , A0 R),
where ξ∗ ∈ ∂∗ D and R > 0 small. We have the following.
Theorem 2. Let D be a John domain with John constant c J and suppose the CDC holds.
Then the following are equivalent:
(i) D is an inner semi-uniform domain.
(ii) (6) holds whenever ξ∗ ∈ ∂∗ D and R > 0 is small.
(iii) For each α > 1/c J , there exist constants A > 1 and τ > 0 depending only on D
and α such that
(
)τ
R
1
∗
for ρD (x, ξ∗ ) < αδD (x),
ω(x, ∆ρ (ξ , R), D) ≥
∗
A R + ρD (x, ξ )
4
HIROAKI AIKAWA AND KENTARO HIRATA
whenever ξ∗ ∈ ∂∗ D and R > 0 is small.
By definition, a semi-uniform domain is an inner semi-uniform domain. The domain in
Figure 1 is an inner semi-uniform domain and satisfies (6). Thus (3) is refined as follows:
NTA $ Uniform
$
$
Inner uniform
$
Inner semi-uniform $ John.
Semi-uniform
$
There is no direct relationship between the class of inner uniform domains and the class of
semi-uniform domains. Theorem 2 and the above implications yield that (4) is a property
stronger than (6). This is not straightforward from their definitions.
The plan of the present paper is as follows: In Section 2, some preliminary notions such
as the quasihyperbolic metric and local reference points will be recalled. The relationship
between the Green function and the harmonic measure will be extensively studied in
Section 3. Theorem 1 will be proved in Section 4 based on the results in Section 3.
Theorem 2 can be proved almost in the same manner. Necessary lemmas will be stated in
the last section.
2. Preliminaries
We define the quasihyperbolic metric kD (x, y) by
∫
ds(z)
kD (x, y) = inf
,
γ
γ δD (z)
where the infimum is taken over all rectifiable curves γ connecting x to y in D. We
observe that the shortest length of the Harnack chain connecting x and y is comparable
to kD (x, y) + 1. Therefore, the Harnack inequality yields that there is a constant A > 1
depending only on n such that
h(x)
(7)
exp(−A(kD (x, y) + 1)) ≤
≤ exp(A(kD (x, y) + 1))
h(y)
for every positive harmonic function h on D. We say that D satisfies a quasihyperbolic
boundary condition if
δD (x0 )
+ A for all x ∈ D.
(8)
kD (x, x0 ) ≤ A log
δD (x)
It is easy to see that a John domain satisfies the quasihyperbolic boundary condition (see
[10, Lemma 3.11]). We have more precise estimate ([3, Proposition 2.1]).
Lemma A. Let D be a John domain with John constant c J . Then there exist a positive
integer N and constants RD > 0 and A > 1 depending only on D with the following
property: for every ξ ∈ ∂D and 0 < R < RD there are N points yR1 , . . . , yRN ∈ D ∩ S (ξ, R)
such that A−1 R ≤ δD (yRi ) ≤ R for i = 1, . . . , N and
R
+ A for x ∈ D ∩ B(ξ, R/2),
δD (x)
where DR = D ∩ B(ξ, 8R). Moreover, every x ∈ D ∩ B(ξ, R/2) can be connected to some
yRi by a curve γ ⊂ DR with ℓ(γ(x, z)) ≤ AδD (z) for all z ∈ γ.
min {kDR (x, yRi )} ≤ A log
i=1,...,N
DOUBLING CONDITIONS FOR HARMONIC MEASURE
5
If the conclusion of the above lemma holds, then we say that ξ has a system of local
reference points yR1 , . . . , yRN of order N.
3. Green function and harmonic measure
We begin by recalling the capacity density condition (abbreviated to CDC).
Definition 3. By Cap we denote the logarithmic capacity if n = 2, and the Newtonian
capacity if n ≥ 3. We say that the CDC holds if there exist constants A > 0 and RD > 0
such that



if n = 2,
AR
Cap(B(ξ, R) \ D) ≥ 

ARn−2 if n ≥ 3,
whenever ξ ∈ ∂D and 0 < R < RD .
It is well known that the CDC is equivalent to the uniformly ∆-regularity ([5]). Hence
there is a positive constant β such that if ξ ∈ ∂D and 0 < r < R are small, then
(9)
sup ω(·, D ∩ S (ξ, R), D ∩ B(ξ, R)) ≤ A(r/R)β ,
D∩B(ξ,r)
so that there is a constant A1 > 1 such that
(10)
1
ω(·, ∂D ∩ B(ξ, R), D) ≥ .
D∩B(ξ,R/A1 )
2
inf
Lemma 1. Let G(x, y) be the Green function for D with the CDC. Suppose δD (y) = R > 0
is small. Then
(11)
G(x, y) ≈ R2−n
for x ∈ S (y, R/2).
Moreover, there is a positive constant β such that
( δD (x) )β
(12)
G(x, y) ≤ AR2−n
for x ∈ D \ B(y, R/2).
R
Proof. If n ≥ 3, then the first assertion is obvious. The planar case will be given in Lemma
3. For the proof of (12) we may assume that δD (x) < R/4. Let x∗ ∈ ∂D be a point such
that |x∗ − x| = δD (x) < R/4. Then |x∗ − y| ≥ δD (y) = R. Hence B(x∗ , R/2) ∩ B(y, R/2) = ∅,
so that the maximum principle and (11) yield
G(x, y) ≤ AR2−n ω(x, S (y, R/2), D\B(y, R/2)) ≤ AR2−n ω(x, D∩S (x∗ , R/2), D∩B(x∗ , R/2)).
Hence we have (12) from (9).
Lemma 2. Let G(x, y) be the Green function for D with the CDC. Suppose δD (y) = R > 0
is small and G(x, y) > A2 R2−n . Then there is a curve γ connecting x and y in D such that
ℓ(γ) ≤ AR and δD (z) ≥ R/A for all z ∈ γ, where A depends only on D and A2 .
Proof. Observe from the maximum principle that Ω = {z ∈ D : G(z, y) > A2 R2−n } is a
connected open set. If n ≥ 3, then G(z, y) ≤ |z − y|2−n , so that diam Ω ≤ AR. The planar
case will be given in Lemma 3. Let γ be a curve connecting x and y in Ω. Lemma 1 says
that
( δD (z) )β
A2 R2−n < G(z, y) ≤ AR2−n
for z ∈ Ω \ B(y, δD (y)/2).
R
Hence δD (z) ≥ R/A for all z ∈ γ. Since diam γ ≤ diam Ω ≤ AR, taking a polygonal curve,
we may modify γ so that ℓ(γ) ≤ AR. The proof is complete.
6
HIROAKI AIKAWA AND KENTARO HIRATA
Lemma 3. Let n = 2 and let G(x, y) be the Green function for D with the CDC. Suppose
δD (y) = R > 0 is small. Then the following statements hold:
(i) G(x, y) ≈ 1 for x ∈ S (y, R/2).
(ii) Let Ω = {z ∈ D : G(z, y) > A2 }. Then diam Ω ≤ AR.
Proof. (i) Let M0 = supS (y,R/2) G(·, y). By the maximum principle G(·, y) ≤ M0 on D \
B(y, R/2). Let y∗ ∈ ∂D be a point such that |y∗ − y| = δD (y) = R. By (9) we find a positive
constant ε1 < 1/4 such that G(·, y) ≤ M0 /2 on D ∩ B(y∗ , 2ε1 R). Let y′ be the point in yy∗
with |y′ − y∗ | = ε1 R. Then G(·, y) ≤ M0 /2 on B(y′ , ε1 R). Cover the sphere S (y, (1 − ε1 )R)
with finitely many balls with the same radii ε1 R. We may assume that B(y′ , ε1 R) appears
in the covering, consecutive balls have an intersection with volume comparable to (ε1 R)n ,
and the number of balls is bounded by a constant depending only on ε1 and the dimension
n. Applying the mean value property of G(·, y), we can conclude G(·, y) ≤ (1 − c)M0 on
S (y, (1 − ε1 )R), and hence on D \ B(y, (1 − ε1 )R) with 0 < c < 1 independent of R and y
(see [2, Proof of Lemma 2]). Let G B be the Green function for B = B(y, (1 − ε1 )R). Then
bD\B (x) ≥ G(x, y) − (1 − c)M0
G B (x, y) = G(x, y) − R
G(·,y)
for x ∈ B,
bD\B is the regularized reduced function of G(·, y) relative to D \ B in D. Take the
where R
G(·,y)
supremum over S (y, R/2) to obtain
A ≥ M0 − (1 − c)M0 = cM0 .
Thus (i) follows, since G(x, y) ≥ G B(y,R) (x, y) = log 2 for x ∈ S (y, R/2).
(ii) For the proof it is sufficient to show the following claim: there is a positive constant
λ such that if δD (y) ≤ 2|x − y| small, then
( δD (y) )λ
(13)
G(x, y) ≤ A
.
|x − y|
Let |x − y| = L be sufficiently small. The first named author ([2, Lemma 1]) showed the
uniform perfectness of ∂D. Hence we find a constant b ≥ 2 and an increasing sequence
δD (y) = R = R1 < R2 < · · · < Rk−1 < L ≤ Rk such that S (y, R j ) ∩ ∂D , ∅ and that
2 ≤ R j /R j−1 ≤ b for j = 1, . . . , k. Here R0 = δD (y)/2. Let u = G(·, y) in D and let
u = 0 in Rn \ D. Then u is a nonnegative subharmonic function in Rn \ {y}. We employ an
argument similar to (i). Cover the sphere S (y, R j ) with finitely many balls with the same
radii ε1 R j . We find y′′ ∈ S (y, R j ) ∩ ∂D. We may assume that B(y′′ , ε1 R j ) appears in the
covering, consecutive balls have an intersection with volume comparable to (ε1 R j )n , and
the number of balls is bounded by a constant depending only on ε1 and the dimension
n. Moreover, observe that these balls lie outside B(y, R j−1 ). Applying the mean value
property of u, we obtain
Mj =
sup u = sup u ≤ (1 − c)M j−1 ≤ (1 − c) j M0
Rn \B(y,R
j)
S (y,R j )
for j = 1, 2, . . . , k. Since L ≤ Rk ≤ bk R0 , it follows that
( R )λ
(
L)
log(1 − c)
log
= M0
Mk ≤ exp(k log(1 − c) + log M0 ) ≤ exp log M0 +
log b
R0
2L
with λ = − log(1 − c)/ log b. Thus (13) follows.
DOUBLING CONDITIONS FOR HARMONIC MEASURE
7
Lemma 4. Let D be a John domain with the CDC. Let ξ ∈ ∂D have a system of local
reference points yR1 , . . . , yRN ∈ D ∩ S (ξ, R) of order N for 0 < R < RD . Then
(14)
R
n−2
N
∑
G(x, yRi ) ≤ Aω(x, ∂D ∩ B(ξ, 2A1 R), D) for x ∈ D \ B(ξ, 2R),
i=1
where A depends only on D and A1 is the constant in (10).
Proof. The maximum principle and (11) give
R
Since
n−2
N
∑
G(x, yRi ) ≈ 1
for x ∈
i=1
∪
i
∪
i
S (yRi , δD (yRi )/2).
S (yRi , δD (yRi )/2) ⊂ D ∩ B(ξ, 2R), it follows from (10) that
ω(x, ∂D ∩ B(ξ, 2A1 R), D) ≈ 1
for x ∈
∪
i
S (yRi , δD (yRi )/2).
The maximum principle completes the proof.
The following is an estimate opposite to Lemma 4.
Lemma 5. Let D be a John domain. Let ξ ∈ ∂D have a system of local reference points
yR1 , . . . , yRN ∈ D ∩ S (ξ, R) of order N for 0 < R < RD . Then
(15)
ω(x, ∂D ∩ B(ξ, R/8), D) ≤ AR
n−2
N
∑
G(x, yRi )
for x ∈ D \ B(ξ, R/4),
i=1
where A depends only on D.
Proof. For 0 < r < δD (x0 )/2 let U(r) = {x ∈ D : δD (x) < r}. Then each point x ∈ U(r)
can be connected to x0 by a curve such that (1) holds. Hence, B(x, A3 r) \ U(r) includes a
ball with radius r, provided A3 is large. This implies that
ω(x, U(r) ∩ S (x, A3 r), U(r) ∩ B(x, A3 r)) ≤ 1 − ε0
for x ∈ U(r)
with 0 < ε0 < 1 depending only on A3 and the dimension. Let R ≥ r and repeat this
argument with the maximum principle. Then
(
R)
(16)
for x ∈ U(r)
ω(x, U(r) ∩ S (x, R), U(r) ∩ B(x, R)) ≤ A exp − A′
r
for some A′ > 0. See [1, Lemma 1] for details.
Let 0 < R < RD . For each x ∈ D ∩ B(ξ, R/2) there is a local reference point y(x) ∈
R
{y1 , . . . , yRN } such that
R
+A
kD (x, y(x)) ≤ A log
δD (x)
by Lemma A. Let y′ (x) ∈ S (y(x), δD (y(x))/2). Observe that kD\{y(x)} (x, y′ (x)) ≤ A log(R/δD (x))+
∑N
A. Letting u(x) = Rn−2 i=1
G(x, yRi ), we obtain from (7) and (11) that
( δD (x) )λ
u(x) ≥ A
for x ∈ D ∩ B(ξ, R/2)
R
with some λ > 0 depending only on D.
8
HIROAKI AIKAWA AND KENTARO HIRATA
Now let us employ a modified version of the box argument (cf. [8] and [1, Lemma 2]).
Let D j = {x ∈ D : exp(−2 j+1 ) ≤ u(x) < exp(−2 j )} and U j = {x ∈ D : u(x) < exp(−2 j )}.
Then we see that
{
( 2 j )}
.
(17)
U j ∩ B(ξ, R/2) ⊂ x ∈ D : δD (x) < AR exp −
λ
Define sequences R j , r j and ρ j by R0 = 3R/8, r0 = R/8 and
ρj =
3 R
,
4π2 j2
∑
3
Rj = R −
ρk ,
8
k=1
j
R ∑
+
ρk
8 k=1
j
rj =
for j ≥ 1. We observe
R
R
3
(18)
= r0 < r1 < · · · < < · · · < R1 < R0 = R.
8
4
8
Let A(ξ, r, R) = B(ξ, R) \ B(ξ, r) be the annulus with center at ξ and radii r and R. Since
R j−1 − R j = r j − r j−1 = ρ j , it follows that if x ∈ A(ξ, r j , R j ), then B(x, ρ j ) ⊂ A(ξ, r j−1 , R j−1 ).
See Figure 2.
Dj
∂D
Dj
ξ
r0
r1
Dj
R1
R
4
R0
∂D
Dj
Figure 2. A box argument for annuli.
The maximum principle, (16) and (17) give
ω(x,U j ∩ ∂A(ξ, r j−1 , R j−1 ), U j ∩ A(ξ, r j−1 , R j−1 ))
(
( 2 j ))
(19)
≤ ω(x, U j ∩ S (x, ρ j ), U j ∩ B(x, ρ j )) ≤ A exp − A j−2 exp
λ
for x ∈ U j ∩ A(ξ, r j , R j ). Let ω0 = ω(·, ∂D ∩ B(ξ, R/8), D) and put

ω0 (x)



if D j ∩ A(ξ, r j , R j ) , ∅,
sup




 x∈D j ∩A(ξ,r j ,R j ) u(x)
dj = 






0
if D j ∩ A(ξ, r j , R j ) = ∅.
By (18) it is sufficient to show that d j is bounded by a constant independent of R and j.
Apply the maximum principle to U j ∩ A(ξ, r j−1 , R j−1 ) to obtain
ω0 (x) ≤ ω(x, U j ∩ ∂A(ξ, r j−1 , R j−1 ), U j ∩ A(ξ, r j−1 , R j−1 )) + d j−1 u(x).
DOUBLING CONDITIONS FOR HARMONIC MEASURE
9
Divide the both sides by u(x) and take the supremum over D j ∩ A(ξ, r j , R j ). Then (19)
yields
(
)
d j ≤ A exp 2 j+1 − A j−2 exp(2 j /λ) + d j−1 .
(
)
∑
Since j exp 2 j+1 − A j−2 exp(2 j /λ) < ∞, we obtain sup j≥0 d j < ∞. Thus (15) follows
from the maximum principle.
4. Proof of Theorem 1
Proof of Theorem 1. (i) =⇒ (ii). Suppose first D is a semi-uniform domain. Let ξ ∈ ∂D
and let R > 0 be sufficiently small. Then by Lemma 5 and scaling we find a system of
local reference points y1 , . . . , yN ∈ D ∩ S (ξ, 16R) such that
ω(x, ∂D ∩ B(ξ, 2R), D) ≤ AR
n−2
N
∑
G(x, yi )
for x ∈ D \ B(ξ, 4R).
i=1
Let {y∗1 , . . . , y∗N } ⊂ D∩S (ξ, R/2A1 ) be a system of local reference points. Lemma 4 implies
that
N
∑
n−2
G(x, y∗i ) ≤ Aω(x, ∂D ∩ B(ξ, R), D) for x ∈ D \ B(ξ, R/A1 ).
R
i=1
By the semi-uniformity, each yi is connected to ξ by a cigar curve γi . Let y′i ∈ γi ∩
S (ξ, R/4A1 ). Observe kD (y′i , y∗j ) ≤ A for some j. Since kD (yi , y∗j ) ≤ kD (yi , y′i ) + kD (y′i , y∗j ) ≤
A and yi , y∗j , y′i ∈ D ∩ B(ξ, 16R), it follows that
G(x, yi ) ≈ G(x, y∗j ) for x ∈ D \ B(ξ, 32R),
so that
ω(x, ∂D ∩ B(ξ, 2R), D) ≤ Aω(x, ∂D ∩ B(ξ, R), D)
for x ∈ D \ B(ξ, 32R).
Hence (4) follows with A0 = 32.
(ii) =⇒ (iii). Suppose ξ ∈ ∂D and R > 0 is small and |x − ξ| < αδD (x). It is easy
to see from (10) that (5) holds for |x − ξ| ≤ R/A1 . Now let r = |x − ξ| > R/A1 . Suppose
first A0 r > RS D with RS D for (4). Take y ∈ D ∩ S (ξ, R/A1 ) with δD (y) ≥ R/A. Then
kD (x, y) ≤ A log(1/R) + A, so that (7) and (10) give
1 τ
1 τ
R ω(y, ∂D ∩ B(ξ, R), D) ≥
R
A
2A
with some τ > 0 depending only on D and α. Since R + |x − ξ| ≥ RS D /A0 , we obtain (5).
Suppose next A0 r ≤ RS D . We find a local reference point yi ∈ D ∩ S (ξ, A0 A1 r) such that
ω(x, ∂D ∩ B(ξ, R), D) ≥
(20)
kD (x, yi ) ≤ A(D, α).
Note that R < A1 r. Applying (4) with yi in please of x repeatedly, we obtain
( r )τ
ω(yi , ∂D ∩ B(ξ, A1 r), D) ≤ A
ω(yi , ∂D ∩ B(ξ, R), D),
R
where A and τ depend only on A1 and the doubling constant. Therefore (7) and (20) give
( r )τ
ω(x, ∂D ∩ B(ξ, A1 r), D) ≤ A
ω(x, ∂D ∩ B(ξ, R), D).
R
10
HIROAKI AIKAWA AND KENTARO HIRATA
Since ω(x, ∂D ∩ B(ξ, A1 r), D) ≥ 1/2 by (10), we obtain (5) as
( R )τ (
)τ
R
≥
.
r
R + |x − ξ|
(iii) =⇒ (i). Let x ∈ D and ξ ∈ ∂D. We may assume that |x − ξ| = R is small. Then by
Lemma A and scaling we find a system of local reference points yR1 , . . . , yRN ∈ D ∩ S (ξ, R)
2R
2R
R
and y2R
1 , . . . , yN ∈ D ∩ S (ξ, 2R). We claim that every yi can be connected to some y j by
a curve γ with ℓ(γ) ≤ AR and δD (z) ≥ R/A for all z ∈ γ. By (iii) and Lemma 5,
∑
1
n−2
R
≤ ω(y2R
,
∂D
∩
B(ξ,
R/8),
D)
≤
AR
G(y2R
i
i , y j ).
A
j=1
N
R
2−n
. Lemma 2 gives a curve γ connecting y2R
Hence there is yRj such that G(y2R
i , y j ) ≥ AR
i
to yRj in D such that ℓ(γ) ≤ AR and δD (z) ≥ R/A for all z ∈ γ. Thus the claim follows.
Now the proof is easy. By Lemma A we find a point y2R
i which can be connected to
x by a cigar curve with length bounded by AR. The claim gives a point yRj which can be
connected to y2R
i by a cigar curve with length bounded by AR. See Figure 3.
y2R
i
yRj
x
∂D
ξ
Figure 3. A cigar curve connecting x to ξ.
Repeat the claim again. We find a point yR/2
which can be connected to yRj by a cigar
k
curve with length bounded by AR/2. Thus we can construct a cigar curve connecting
points as follows:
R/2
R
→ · · · → ξ.
x → y2R
i → y j → yk
The length of the curve is bounded by AR. Thus D is a semi-uniform domain.
5. Proof of Theorem 2
Replacing Lemmas A, 4 and 5 by the following three lemmas, we can prove Theorem
2 almost in the same way as for Theorem 1. The details are left to the reader. Recall π
is the natural projection from D∗ to D. Let ξ∗ ∈ ∂∗ D, ξ = π(ξ∗ ) and S ρ (ξ∗ , R) = {x ∈
D : ρD (x, ξ∗ ) = R}. Observe that S ρ (ξ∗ , R) ⊂ S (ξ, R), that Bρ (ξ∗ , R) is the connected
component of B(ξ, R) ∩ D from which ξ∗ is accessible, and that the boundary of Bρ (ξ∗ , R)
is included in S ρ (ξ∗ , R) ∪ ∂D. The following lemma corresponds to Lemma A.
DOUBLING CONDITIONS FOR HARMONIC MEASURE
11
Lemma 6. Let D be a John domain with John constant c J . Then there exist a positive
integer M and constants RD > 0 and A > 1 depending only on D with the following
property: for every ξ∗ ∈ ∂∗ D and 0 < R < RD there are M points yR1 , . . . , yRM ∈ S ρ (ξ∗ , R)
such that A−1 R ≤ δD (yRi ) ≤ R for i = 1, . . . , M and
min {kBρ (ξ∗ ,8R) (x, yRi )} ≤ A log
i=1,...,M
R
+ A for x ∈ Bρ (ξ∗ , R/2).
δD (x)
Moreover, every x ∈ Bρ (ξ∗ , R/2) can be connected to some yRi by a curve γ ⊂ Bρ (ξ∗ , 8R)
with ℓ(γ(x, z)) ≤ AδD (z) for all z ∈ γ.
If the conclusion of the above lemma holds, then we say that ξ∗ ∈ ∂∗ D has a system
of inner local reference points yR1 , . . . , yRM of order M. We emphasize that inner local
reference points yR1 , . . . , yRM lie on S ρ (ξ∗ , R) and that M ≤ N in general. The following two
lemmas replace Lemmas 4 and 5.
Lemma 7. Let D be a John domain with the CDC. Let ξ∗ ∈ ∂∗ D have a system of inner
local reference points yR1 , . . . , yRM ∈ S ρ (ξ∗ , R) of order M. Then
R
n−2
M
∑
G(x, yRi ) ≤ Aω(x, ∆ρ (ξ∗ , 2A1 R), D) for x ∈ D \ Bρ (ξ∗ , 2R),
i=1
where A depends only on D.
Lemma 8. Let D be a John domain. Let ξ∗ ∈ ∂∗ D have a system of inner local reference
points yR1 , . . . , yRM ∈ S ρ (ξ∗ , R) of order M. Then
∗
ω(x, ∆ρ (ξ , R/8), D) ≤ AR
n−2
M
∑
G(x, yRi )
for x ∈ D \ Bρ (ξ∗ , R/4),
i=1
where A depends only on D.
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Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
E-mail address: [email protected]
Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
E-mail address: [email protected]